Quantum Mutual Information Capacity for HighDimensional Entangled States
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Dixon, P. et al. “Quantum Mutual Information Capacity for HighDimensional Entangled States.” Physical Review Letters 108.14
(2012). © 2012 American Physical Society
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http://dx.doi.org/10.1103/PhysRevLett.108.143603
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American Physical Society
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PRL 108, 143603 (2012)
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6 APRIL 2012
PHYSICAL REVIEW LETTERS
Quantum Mutual Information Capacity for High-Dimensional Entangled States
P. Ben Dixon,* Gregory A. Howland, James Schneeloch, and John C. Howell
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
(Received 5 August 2011; revised manuscript received 2 February 2012; published 5 April 2012)
High-dimensional Hilbert spaces used for quantum communication channels offer the possibility of
large data transmission capabilities. We propose a method of characterizing the channel capacity of an
entangled photonic state in high-dimensional position and momentum bases. We use this method to
measure the channel capacity of a parametric down-conversion state by measuring in up to 576
dimensions per detector. We achieve a channel capacity over 7 bits=photon in either the position or
momentum basis. Furthermore, we provide a correspondingly high-dimensional separability bound that
suggests that the channel performance cannot be replicated classically.
DOI: 10.1103/PhysRevLett.108.143603
PACS numbers: 42.50.Dv, 03.67.Bg, 03.67.Hk
Typical examples of quantum entanglement involve
two-dimensional bipartite systems [1,2]; however, quantum systems can exhibit more complex forms of entanglement, and attention has recently been turned to studying
these rich forms of entanglement. This includes research
on entanglement in multiple degrees of freedom (known as
‘‘hyperentanglement’’) [3,4], as well as high-dimensional
systems entangled in a single degree of freedom. These
degrees of freedom include time and energy, and
transverse-position and momentum (including angular
momentum) [5–10]. Both of these types of entanglement
can present advantages for practical applications. Indeed,
for communication purposes—such as quantum key distribution [11] or dense coding [12]—higher dimensional
states increase quantum channel capacity and offer additional benefits such as increased security [13,14].
The photonic transverse-position degree of freedom is a
good candidate for practical high-dimensional entangled
systems due to the wide availability of technology for
manipulating this degree of freedom. As a result there
has been significant theoretical and experimental effort to
increase and characterize the channel capacity of entangled
photons in this transverse domain. Pors et al. demonstrated
a measurement-limited Shannon dimensionality (the effective number of measurable modes, see [15]) of D 6
using angular phase plates for a system with a Schmidt
number of K 31 [16]. Dada et al. demonstrated a violation of an 11-dimensional Bell inequality using spatial
light modulators as mode sorters for orbital angular
momentum states [17].
These examples show how previous characterization
schemes rely on reconstructing continuous transformation
properties of the state, which are in general determined
either by direct measurement or by reconstructing the
density matrix. The reconstruction is then used to calculate
the corresponding channel capacity. These methods, however, are difficult to scale to arbitrary dimensions; either
the analyzer fabrication requirements exceed the state of
the art (e.g., Pors et al. anticipated a future Shannon
0031-9007=12=108(14)=143603(5)
dimensionality realization of D 50), the crosstalk
between mode decompositions becomes too large, or
the calculations involve very demanding maximization
procedures.
Here we propose a direct way of characterizing the
communication capabilities of a high-dimensional quantum system—entangled in the photonic transverse-position
degree of freedom—that offers several advantages over
other methods. This method does not have the intermediate
step of determining the transformation properties of the
state, but rather uses the knowledge of mutually unbiased
bases for our quantum system (the position basis and the
momentum basis). This allows us to characterize the system by only measuring in these two bases, drastically
reducing the number of required measurements and, in
the process, bypassing many of the scaling problems of
previous methods. Additionally, this method has the practical benefit of using the communication apparatus itself
for the characterization, thus simplifying system requirements and more directly linking the characterization to the
system’s ultimate communication capabilities.
We demonstrate the utility of this method on a parametric down-conversion state with a Schmidt number in
excess of 1000. Employing classical definitions of mutual
information for joint photon detections, and measuring in
up to 576 dimensions per detector, we show that the system
achieved a channel capacity of over 7 bits per joint photon
detection event, roughly equivalent to a dimensionality of
128. We show that this high-dimensional entangled system
relies upon quantum correlations by violating the correspondingly high-dimensional classical separability bound.
We also show that for Gaussian correlated states in the lownoise limit and as the number of detectors becomes large,
the mutual information characterization asymptotically
approaches the Schmidt number of the entangled bipartite
state.
We characterize our channel using the concept of mutual
information, which describes how much information can
be determined about a random variable A, by knowing the
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Ó 2012 American Physical Society
PHYSICAL REVIEW LETTERS
PRL 108, 143603 (2012)
value of a correlated random variable B [18,19]. Variables
A and B are characterized by the values they take a and b,
respectively, and the probability of these values pðaÞ
and pðbÞ, respectively. The mutual information can be
written as
IðA; BÞ ¼ HðAÞ þ HðBÞ HðA; BÞ;
where, for example,
HðAÞ ¼ X
pðaÞ logpðaÞ
is the marginal entropy of A, and
X
HðA; BÞ ¼ pða; bÞ logpða; bÞ
(3)
is the joint entropy of A and B. The function pða; bÞ is the
joint probability distribution that characterizes the correlation between A and B.
We created a position-momentum entangled state using
spontaneous parametric down-conversion (SPDC). The
state, represented both in position and in momentum, is
approximated as [20]
Z
j c i ¼ dx~ a dx~ b fðx~ a ; x~ b Þa^ ya a^ yb j0i
Z
~ k~a ; k~b Þa^ ya a^ y j0i;
dk~a dk~b fð
b
is the approximated entangled biphoton wave function in
the position basis, and
~ k~a ; k~b Þ ¼ ð4p c Þ2 N expð 2c ðk~a k~b Þ2 Þ
fð
expð 42p ðk~a þ k~b Þ2 Þ
(2)
a2A
b2B
¼
ðx~ a x~ b Þ2
ðx~ a þ x~ b Þ2
exp
(5)
42c
162p
fðx~ a ; x~ b Þ ¼ N exp
(1)
a2A
(4)
where a^ y is the photon creation operator. Subscript a
indicates the photon is created in the signal mode and
subscript b indicates the photon is created in the idler
mode, which are then sent to Alice or Bob, respectively.
The function
FIG. 1 (color online). Experimental setup. A collimated laser
beam undergoes spontaneous parametric down-conversion at a
nonlinear crystal. The output passes a focusing lens followed by
a beam splitter. The outputs from the beam splitter are sent to
digital micromirror devices at either image planes or Fourier
planes of the crystal. The micromirror devices are set to retroreflect the beams, a quarter wave plate and a polarizing beam
splitter send the retroreflected beam to a single-photon detector.
A coincidence circuit correlates these measurements.
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(6)
is the approximated biphoton wave function in the momentum basis. In these equations p is the Gaussian width
in the x~ a þ x~ b direction and controls the single-photon
width: c is the Gaussian width in the x~ a x~ b direction
and controls the two photon correlation width. N ¼
ð2p c Þ1 is a normalization constant.
To
measure
position
correlations
we
put
spatially resolving single-photon detectors at image planes
of the SPDC source: to measure momentum correlations
we put the detectors at Fourier transform planes of the
source. For our purposes, then, random variable A corresponds either to the position or momentum of Alice’s
photon and B corresponds either to the position or momentum of Bob’s photon.
The theoretical maximum mutual information for the
wave function in Eq. (5) (measuring in the position basis)
Z
pðx~ a ; x~ b Þ
IðA; BÞ ¼ pðx~ a ; x~ b Þ log
dx~ a dx~ b ; (7)
pðx~a Þpðx~ b Þ
and
pðx~ a Þ ¼
where
pðx~ a ; x~ i Þ ¼ jfðx~ a ; x~ b Þj2
R
jfðx~ a ; x~ b Þj2 dx~ b . For the momentum basis, the same relations hold, but the position variables are replaced by the
momentum variables and the position wave function is
replaced by the momentum wave function of Eq. (6).
For either basis, this theoretical maximum simplifies to
2
4p þ 2c 2
IðA; BÞ ¼ log
;
(8)
4c p
which is independent of detector characteristics. In the
limit of strong correlations ðp =c Þ 1, the mutual
information reduces to IðA; BÞ ¼ logðp =c Þ2 . The ratio
p =c is the familiar Fedorov ratio for quantifying entanglement—which is identical to the Schmidt number for
Gaussian entangled states [21]. Our physical SPDC state
had a beam envelope width of p ¼ 1500 m and a
correlation width of approximately c ¼ 40 m, resulting
in an optimum mutual information from Eq. (8) of I ffi
10 bits=photon.
It should be noted that although we use pure quantum
states, mutual information characterization can be applied
to more general states including mixed states or bipartite
multiparticle systems. For these situations, Eq. (7) would
be valid but the probability distributions would be calculated in a different manner. Additionally, although we
may be sacrificing some intuitive advantages by using an
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entropic measure rather than dimensionality, we do gain a
more direct link to information theory.
The measurement apparatus (shown in Fig. 1) consisted
of a digital micromirror device (DMD) chip reflecting a
portion of the signal or idler beam onto a single-photon
counting module. The DMD chip allowed us to raster scan
over the face of the beam in a controllable number of
detection pixels, giving varying detector resolution. To
incorporate the effects of the measurement apparatus, we
integrate the probability density over the pixel area. For
example, for position correlations between the mth pixel
on Alice’s detector, and the nth pixel on Bob’s detector, the
joint detection probability is
Z
Z
pðm; nÞ ¼
dx~ a dx~ b jfðx~ a ; x~ b Þj2 :
(9)
m
n
The detected mutual information in either position or
momentum is
X
X
IðA; BÞ ¼ pðmÞ logpðmÞ pðnÞ logpðnÞ
m
n
X
þ pðm; nÞ logpðm; nÞ;
(10)
m;n
where, for example,
pðmÞ ¼
X
pðm; nÞ
(11)
n
is the marginal probability for a pixel on Alice’s detector.
With ideal alignment, a given pixel on Alice’s detector
will strongly correlate to only one pixel on Bob’s detector. In practice, however, a relative lateral shift of pixels
between Alice’s and Bob’s detectors—both vertically and
horizontally—spreads correlations to four pixels at best.
However, pixels far from the correlated pixel will still have
no correlation. This was verified experimentally and it
allowed us, for a given pixel on Alice’s detector, to scan
Mutual Information bits photon
Mutual Information bits photon
Position Correlations
10
8
6
4
2
0
only in a region of interest around the correlated pixel on
Bob’s detector, thus reducing the time required to complete
a double raster scan.
Joint photon detection rates for the various raster
scan measurements were between 1 and 100 per second
(between correlated pixels). Integration time for raster
scans was increased as the count rates decreased, with
the longest time being 5 s per pixel pair. Multiple scans
were performed for each detector resolution.
Both the predicted and experimentally measured values
for mutual information are shown in Fig. 2. Mutual information values for both position correlation measurements
and momentum correlation measurements are presented.
For an accurate characterization, only the joint photon
detection events are used to determine the mutual information. Single detection events without the corresponding
detection in the other arm are ignored even for the
calculation of the marginal probabilities of Eq. (11).
Uncertainties in the number of detected photons N at
each
pffiffiffiffi point in the double raster scan were assumed to be
N . This was used to find the uncertainties of the measured
mutual information values, which agree with the statistics
found by taking multiple data scans for a given detector
resolution. It should be noted that this uncertainty calculation method does not take into account detector dark
counts. Since the dark counts from each detector are uncorrelated, the dark coincident rate is much less than the
coincident rate from the highly correlated SPDC state.
Light blue bars represent predicted mutual information
values from numerical calculations of Eq. (10). The top of
each bar corresponds to perfect lateral pixel alignment
between Alice and Bob, and the bottom corresponds to
relative lateral shifts, both horizontally and vertically, of
half a pixel. These cases represent the maximum and
minimum mutual information possible for a given number
of detector pixels. The dark blue circles represent
Momentum Correlations
10
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PHYSICAL REVIEW LETTERS
PRL 108, 143603 (2012)
0
5
10
15
20
25
Detector Resolution pixels
30
8
6
4
2
0
0
5
10
15
20
25
Detector Resolution pixels
30
FIG. 2 (color online). Mutual information data for position correlation measurements and momentum correlation measurements are
shown as a function of detector resolution. Data for detector resolutions of 8 8 pixels, 16 16 pixels, and 24 24 pixels are shown.
The dark blue points with error bars are experimental data. The light blue bars are numerical simulations based on Eq. (10) both for the
case of perfectly relative transversely aligned detectors and the case of a relative transverse misalignment of half a pixel. The red curve
is the maximum mutual information that can be detected for the number pixels per detector.
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PRL 108, 143603 (2012)
PHYSICAL REVIEW LETTERS
experimentally measured channel capacities. The red curve
gives the maximum mutual information that can be
detected I ¼ logðn nÞ for n n pixels per detector
(independent of state). The experimental data agree with
the theoretically predicted values, staying within the
predicted range for all data sets.
For momentum correlations a maximum mutual information of 7:2 0:3 bits=photon was achieved; for position
correlations only 7:1 0:7 bits=photon were achieved. In
principle, the two measurement bases should give the same
mutual information. However, the alignment for position
correlations was more sensitive—the reduction of mutual
information for this basis most likely resulted from slight
system misalignments.
The 576 dimensional measurement space is 16 times
larger than the previous maximum for position-momentum
entangled photons, and over 50 times larger than recent
realizations using the photonic orbital angular momentum
degree of freedom [17]. It should be noted that channel
capacity characterization is different from using the channel for communication. When used for key distribution or
communication, the characterized channel will indeed
transmit 7 bits of information for a single joint detection
event, despite the fact that characterization methods
require many photon detection events. The use of this
channel for key distribution or communication does, however, require some additional structure [11,12], and we are
further investigating the ultimate experimental realization
of these structures. It should be noted, however, that
although we only detect photons from one pixel at a
time, photons from the other pixels are not necessarily
lost—rather, they are reflected to a different (but known)
location that could in principle be monitored. This could be
important in requirements for quantum information protocols, for example, in the detection of eavesdropping.
Recent work suggests that this data can be used to test
for nonclassical behavior of the communication channel. A
separable quantum state in two spatial dimensions satisfies
the inequality hðAjBÞP þ hðAjBÞM 2log2 ðeÞ, where,
for example, hðAÞ is the continuous (or differential) entropy of A, hðAjBÞ ¼ hðA; BÞ hðBÞ is the conditional
continuous entropy of A given B, and subscripts P or M
indicate measurements in the position or momentum bases,
respectively [19,22]. Although the inequality is derived
using continuous entropies, Walborn et al. have made use
of it with discrete entropies by approximating the relationship between the two entropies [23]. We make use of a
modified inequality where the continuous entropy is
replaced with the discrete entropy:
HðAjBÞP þ HðAjBÞM 2log2 ðeÞ 6:19:
(12)
It should be noted that we have not yet derived a proof of
this form of the inequality and we are currently researching
its full range of applicability.
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From the 24 24 pixel scan data, we calculate
HðAjBÞP þ HðAjBÞM ¼ 2:2 0:7;
(13)
HðBjAÞP þ HðBjAÞM ¼ 2:2 0:6:
(14)
The separability bound of Eq. (12) is violated by more than
5 standard deviations, suggesting that the channel performance cannot be replicated classically.
In this Letter we have proposed and demonstrated a
method of characterizing the quantum mutual information
based channel capacity of a high-dimensional quantum
communication channel using position and momentum
entangled photons and a controllable pixel mirror. We
measured up to 576 dimensions per detector, in both the
position and the momentum basis, which resulted in a
measured channel capacity of more than 7 bits=photon
for either basis. The channel violated an entropic separability bound, strongly suggesting the performance cannot
be replicated classically.
We acknowledge discussions with B. I. Erkmen and
support from DARPA DSO InPho Grant No. W911NF10-1-0404 and USARO MURI Grant No. W911NF-05-10197.
*Present address: Research Laboratory of Electronics,
Massachusetts Institute of Technology, Cambridge, MA
02139, USA.
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